Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Bird brain

A tree has a left branch and a right branch, with some birds on both branches. If one bird flies from the left to the right branch, then the numbers of birds would be the same on both branches. However, if one bird flies from the right branch to the left branch, then the number of birds on the left branch would be double the number of birds on the right branch.

2 marks

1.1 Initially, how many birds are on the left branch?

Correct Solution: 7

Initially there must have been 7 birds on the left branch

2 marks

1.2 Initially, how many birds are on the right branch?

Correct Solution: 5

Initially there must have been 5 birds on the right branch

2. Intermediate Maths Challenge Problem (UKMT)

3. Jugs of water – more interesting than it sounds

There are a whole set of mathematical puzzles based on trying to measure out specific amounts of water using jugs, jars or buckets of fixed volume. One such puzzle was featured in the Hollywood action movie “Die Hard with a Vengeance”, starring Bruce Willis and Samuel L. Jackson, as shown in the clip below. In the clip, you are given a chance to pause the video, so that you can try to solve the problem yourself.

If you found that previous video confusing – after all, Hollywood movies are not usually designed to explain mathematics - then it might help to watch this second video.

Next week, we will look at some trickier water jug problems. In the meantime, I will leave you with one of the happiest surprises in my life, when I appeared in Viz magazine with this rather odd jug puzzle.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1. Weighing the baby at the clinic was a problem. The baby would not keep still and caused the scales to wobble. So I held the baby and stood on the scales while the nurse read off 78 kg. Then the nurse held the baby while I read off 69 kg. Finally I held the nurse while the baby read off 137 kg.

What was the combined weight of all three?

142 kg

147 kg

206 kg

215 kg

284 kg

(This problem appeared in the first Schools' Mathematical Challenge in 1988.)

We let the weights of the baby, the nurse and myself be x kg, y kg and z kg, respectively. The information we are given implies that x+z=78, x+y=69 and y+z=137.

Adding these three equations gives x+z+x+y+y+z=78+69+139, that is, 2x+2y+2z=284.

It follows that x+y+z=142. So the combined weight of all three was 142 kg.

5. What is the diagonal?

3 marks

5.1. The diagram shows a quarter of a circle.

Within the circle is the rectangle ABCD, which touches the edge of the circle at point D.

Point B is at the centre of the circle.

How long is the diagonal line AC?

Correct Solution: 10 cm

Show Hint (–1 mark)

–1 mark

Consider the other diagonal, BD.

Diagonal BD is the same length as the diagonal AC, and BD is equal to the radius of the circle. We can see from the horizontal radius that it has length of 5 cm + 5cm = 10 cm.